# What is the arclength of f(x)=(x-3)e^x-xln(x/2) on x in [2,3]?

Jul 13, 2018

$\approx 6.533320817$

#### Explanation:

We are using the Formula

${\int}_{a}^{b} \sqrt{1 + f ' {\left(x\right)}^{2}} \mathrm{dx}$
$f \left(x\right) = \left(x - 3\right) {e}^{x} - x \ln \left(\frac{x}{2}\right)$
we Need the first derivative of $f$

$f ' \left(x\right) = {e}^{x} + {e}^{x} \left(x - 3\right) - \ln \left(\frac{x}{2}\right) - x \cdot \left(\frac{1}{\frac{x}{2}} \cdot \frac{1}{2}\right)$

simplifying we get

$f ' \left(x\right) = {e}^{x} \left(x - 2\right) - \ln \left(\frac{x}{2}\right) - 1$
so we have to calculate

${\int}_{2}^{3} \sqrt{1 + {\left({e}^{x} \left(x - 2\right) - \ln \left(\frac{x}{2}\right) - 1\right)}^{2}} \mathrm{dx}$
by a numerical method we get

$\approx 6.533320817$